two spheres carrying charge q are hanging from a same point of suspension with the help of threads of length 1 m in a space free from gravity . the distance between them is?
To find the distance between the two charged spheres hanging from a common point of suspension, we can analyze the equilibrium condition.
Let's assume that the distance between the centers of the spheres (hanging point to hanging point) is "d" meters.
Since the setup is in a space free from gravity, we can neglect the effect of gravity on the system. Thus, the only forces acting on the charged spheres are due to the charges themselves.
Consider the forces acting on one sphere due to the other:
1. Electrostatic Repulsion: The spheres carry charges of equal magnitude "q". Like charges repel each other, resulting in a repulsive force pushing the spheres apart.
For a sphere of radius "r" and charge "q", the electrostatic force between the two spheres can be calculated using Coulomb's Law:
F = (k * q^2) / (d^2)
Where:
- F is the electrostatic force between the spheres,
- k is the electrostatic constant (approximately 9 × 10^9 Nm^2/C^2),
- q is the magnitude of the charge on each sphere, and
- d is the distance between the centers of the spheres.
Since the spheres are in equilibrium, the electrostatic force is balanced by the tension in the threads. The tension forces on each sphere are equal and act radially towards the center of the suspended point.
2. Tension Forces: The tension in the threads provides the centripetal force required to keep the spheres in circular motion. The tension force in each thread can be represented as T.
We can consider the vertical components of the tension forces, which balance the weight of the charged spheres (since we are in a space free from gravity). The vertical components of the tension forces in each thread are equal to q * g, where g is the acceleration due to gravity (which we can neglect since we are in a gravity-free space).
Now, we can equate the vertical component of the tension forces to the electrostatic force:
T * sin(θ) = F
T * sin(θ) = (k * q^2) / (d^2)
Since the threads are vertical, sin(θ) = 1.
Thus, we have:
T = (k * q^2) / (d^2)
Since the threads are of length 1m, the vertical component of the tension force is also equal to 1 (T = 1).
Now, we can equate these two expressions:
1 = (k * q^2) / (d^2)
Rearranging the equation:
d^2 = (k * q^2)
Taking the square root of both sides:
d = sqrt(k * q^2)
Therefore, the distance between the centers of the spheres is given by "d = sqrt(k * q^2)".
the downward force on each is mg.
from the vertical, then the angle to one string Theta is given by
kQQ/r^2mg=tanTheta
but sinTheta=r/2L
assuming small angles, sinTheta appx=tantheta
r/2L=kQQ/r^2mg
r^3=kQQL/mg
check that